In this section, we will derive the generalised flow equation for quantum gravity from which we use to apply the essential RG method in order to investigate asymptotic safety. This construction generalises the formalism introduced in [1]} by allowing for the field redefinitions at the heart of the essential RG.
For quantum gravity the EAA is denoted \(\Gamma _k[f;\bar{g}]\) , where \(f= \lbrace  g_{\mu \nu }, c^{\mu }, \bar{c}_{\mu } \rbrace \)  denotes the set of mean fields, \(g_{\mu \nu }\)  is the (mean) metric, and \( c^{\mu }\)  and \(\bar{c}_{\mu }\)  are the (mean) anti-commuting ghost and anti-ghost. In addition to the mean fields, \(\Gamma _k[f;\bar{g}]\)  also depends on an auxiliary background metric \(\bar{g}_{\mu \nu }\)  in order to conserve background covariance. The EAA for gravity is defined analogously to the case of the scalar field (REF ) through the functional integral
\({\rm {e}}^{- \Gamma _k[f,\bar{g}]} = \int \left({\rm {d}} \hat{\chi }\right)\, {\rm {e}}^{-S[\hat{\chi }; \bar{g} ]} {\rm {e}}^{(\hat{f}_k[\hat{\chi }] - f)\cdot \frac{\delta }{\delta f} \Gamma _k[f;\bar{g}] }{\rm {e}}^{- \frac{1}{2} (\hat{f}_k[\hat{\chi }] - f)\cdot \mathcal {R}_k[\bar{g}] \cdot (\hat{f}_k[\hat{\chi }] - f) } \,,\) 
